Aerospace Technic and Technology

The relevance of the work is due to the increase in power and the minimization of the weight of modern aircraft, which is associated with an increase in dynamic loads on the units and the development of the nonlinearity of dynamic processes in them. The problem of designing devices and units operating at high dynamic loads can be solved by developing and applying in designing nonlinear models and methods for analyzing nonlinear oscillatory processes. The use of nonlinear models and methods in design and calibration calculations makes it possible to predict dynamic processes in those aircraft operating modes that cannot be achieved using linear models. The research aims to develop models and methods for analyzing nonlinear oscillatory processes for designing equipment and aircraft assemblies. The rotors, in which two paddle wheels mounted on the cantilever ends of the shaft, are most often used in engines, turbo-pump rocket units, and turbo coolers of aircraft. The shaft deflections have the same order as the elastic deformations of the bearings. Approximation of the axis of the deformed shaft of such rotor design is difficult to implement by sinusoidal functions, therefore the finite element method is used. The finite elements approximate the areas of the shaft of the constant cross section. Disks and supports are placed in the nodes. The inertia forces and moments of inertia forces of the disks are considered linear boundary conditions at the nodes of finite elements. The elastic forces of bearings are considered non-linear boundary conditions at the nodes. The interpolation polynomials of these finite elements are functions of the bent axis of the beam with unit displacements of the nodal sections. The oscillation equations of the shaft are obtained by the Galerkin method with a simultaneous approximation of differential equations and boundary conditions. For the analysis of free vibrations, we use the method of nonlinear normal modes (NNM), which allow us to reduce the analysis of a system with a finite number of degrees of freedom to an analysis of an oscillator with one degree of freedom. Following this method all phase coordinates are represented as functions of one pair of phase coordinates, they are a generalized displacement which can be chosen arbitrarily and a corresponding generalized velocity. Elements of these functions are represented by Taylor series. The rotors on angular contact ball bearings with axial preload are investigated in the work. The preload is used to eliminate the opening of the gaps between the balls and the racers during vibrations, which can lead to shocks and increased vibrations. For the convenience of using the NNM method, the elastic forces are represented as power series in generalized coordinates. The equation of oscillations for each NNM is solved by the method of harmonic balance. The shapes and backbone curves of free non-linear oscillations of the rotor are constructed.

flexible shaft; disk; angular contact ball-bearing; nonlinear vibration; nonlinear normal mode; backbone curve

Harris, T. A. Rolling bearing analysis. 4th ed. New York, John Wiley and Sons Inc., 2001. 1086 p.

Tiwari, M., Gupta, K., Prakash, O. Dynamic response of an unbalanced rotor supported on ball bearings. Journal of Sound and Vibration, 2000, vol. 238, iss. 5, pp. 757–779. DOI: 10.1006/jsvi.1999.3109.

Young, T. H., Shiau, T. N., Kuo, Z. H. Dynamic stability of rotor-bearing systems subjected to random axial forces. Journal of Sound and Vibration, 2007, vol. 305, iss. 3, pp. 467–480. DOI: 10.1016/j.jsv.2007.04.016.

Harsha, S. P. Nonlinear dynamic analysis of a high-speed rotor supported by rolling element bearings. Journal of Sound and Vibration, 2006, vol. 290, iss. 1/2, pp. 65–100. DOI: 10.1016/j.jsv.2005.03.008.

Xuening, Z. et al. Stability analysis of a rotor–bearing system with time-varying bearing stiffness due to finite number of balls and unbalanced force. Journal of Sound and Vibration, 2013, vol. 332, iss. 25, pp. 6768–6784. DOI: 10.1016/j.jsv.2013.08.002.

Tomovic, R. Vibration response of rigid rotor in unloaded rolling element bearing. International Journal of Mechanical Sciences, 2010, vol. 52, iss. 9, pp. 1176–1185. DOI: 10.1016/j.ijmecsci.2010.05.003.

Filipkovskii, S. V., Avramov, K. V. Nonlinear Free Vibrations of Multi-Disk Rotors on Ball Bearings. Strength of Materials, 2013, vol. 45(3), pp. 316–323. DOI: 10.1007/s11223-013-9461-2.

Filipkovskiy, S. V. Nonlinear Dynamic Analysis Of Elastic Rotor With Disk On Cantilevered End Supported On Angular Contact Ball Bearings. Proc. of the 5th Intern. Conf. on Nonlinear Dynamics ND-KhPI’2016, Sept. 27–30, 2016, Kharkov, Ukraine. Nat. Techn. Univ. «Kharkov Polytechnic Inst.», Kharkov, 2016, pp. 74–81.